Effect of worst-case partial band jamming

Question

Solve the following problem:

In a coded and interleaved FH q-ary FSK modulation with partial band jamming and coherent demodulation with soft-decision decoding, the cutoff rate is

R₀ = log₂ [q/1 + (q - 1)αe^{-αε_c/2N₀}]

where α is the fraction of the band being jammed, E_c is the chip (or tone) energy, and N₀ = J₀.

a. Show that the SNR per bit can be expressed as

ε_b/N₀ = 2/αR In (q - 1)α/q^2-R₀ -1

b. Determine the value of α that maximizes the required ε_b/N₀ (worst-case partial band jamming) and the resulting maximum value of ε_b/N₀.

c. Define r = R₀/R in the result for ε_b/N₀ from (b), and plot 10 log(ε_b/r N₀) versus the normalized cut off rate R₀/ log₂ q for q = 2, 4, 8, 16, 32. Compare these graphs with the results of Problem A. What conclusions do you reach regarding the effect of worst-case partial band jamming? What is the effect of increasing the alphabet size q?

What is the penalty in SNR between the results in Problem A and q-ary FSK as q → ∞?

Problem A: Plot the graph of 10 log(ε_b/r N₀) versus R0, where r = R₀/R, for worst-case pulse jamming and for AWGN (α = 1). What conclusions do you reach regarding the effect of worst-case pulse jamming?